Find the Inverse y=x/2+3

$y = \frac{x}{2} + 3$

Step 1

Interchange the variables.

$x = \frac{y}{2} + 3$

Step 2

Solve for $y$ .

Rewrite the equation as $\frac{y}{2} + 3 = x$ .

$\frac{y}{2} + 3 = x$

Subtract $3$ from both sides of the equation.

$\frac{y}{2} = x - 3$

Multiply both sides of the equation by $2$ .

$2 \frac{y}{2} = 2 (x - 3)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{y}{2} = 2 (x - 3)$

Rewrite the expression.

$y = 2 (x - 3)$

Simplify the right side.

Simplify $2 (x - 3)$ .

Apply the distributive property.

$y = 2 x + 2 \cdot - 3$

Multiply $2$ by $- 3$ .

$y = 2 x - 6$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = 2 x - 6$

Step 4

Verify if $f^{- 1} (x) = 2 x - 6$ is the inverse of $f (x) = \frac{x}{2} + 3$ .

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Evaluate $f^{- 1} (f (x))$ .

Set up the composite result function.

$f^{- 1} (f (x))$

Evaluate $f^{- 1} (\frac{x}{2} + 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{x}{2} + 3) = 2 (\frac{x}{2} + 3) - 6$

Simplify each term.

Apply the distributive property.

$f^{- 1} (\frac{x}{2} + 3) = 2 (\frac{x}{2}) + 2 \cdot 3 - 6$

Cancel the common factor of $2$ .

Cancel the common factor.

$f^{- 1} (\frac{x}{2} + 3) = 2 (\frac{x}{2}) + 2 \cdot 3 - 6$

Rewrite the expression.

$f^{- 1} (\frac{x}{2} + 3) = x + 2 \cdot 3 - 6$

Multiply $2$ by $3$ .

$f^{- 1} (\frac{x}{2} + 3) = x + 6 - 6$

Combine the opposite terms in $x + 6 - 6$ .

Subtract $6$ from $6$ .

$f^{- 1} (\frac{x}{2} + 3) = x + 0$

Add $x$ and $0$ .

$f^{- 1} (\frac{x}{2} + 3) = x$

Evaluate $f (f^{- 1} (x))$ .

Set up the composite result function.

$f (f^{- 1} (x))$

Evaluate $f (2 x - 6)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (2 x - 6) = \frac{2 x - 6}{2} + 3$

Cancel the common factor of $2 x - 6$ and $2$ .

Factor $2$ out of $2 x$ .

$f (2 x - 6) = \frac{2 (x) - 6}{2} + 3$

Factor $2$ out of $- 6$ .

$f (2 x - 6) = \frac{2 (x) + 2 \cdot - 3}{2} + 3$

Factor $2$ out of $2 (x) + 2 (- 3)$ .

$f (2 x - 6) = \frac{2 (x - 3)}{2} + 3$

Cancel the common factors.

Factor $2$ out of $2$ .

$f (2 x - 6) = \frac{2 (x - 3)}{2 (1)} + 3$

Cancel the common factor.

$f (2 x - 6) = \frac{2 (x - 3)}{2 \cdot 1} + 3$

Rewrite the expression.

$f (2 x - 6) = \frac{x - 3}{1} + 3$

Divide $x - 3$ by $1$ .

$f (2 x - 6) = x - 3 + 3$

Combine the opposite terms in $x - 3 + 3$ .

Add $- 3$ and $3$ .

$f (2 x - 6) = x + 0$

Add $x$ and $0$ .

$f (2 x - 6) = x$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 2 x - 6$ is the inverse of $f (x) = \frac{x}{2} + 3$ .

$f^{- 1} (x) = 2 x - 6$

Find the Inverse y=x/2+3

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